Abstract

A method to construct noncommutative instantons as deformations from commutative instantons was provided by Maeda and Sako [J. Geom. Phys.58, 1784 (2008)]10.1016/j.geomphys.2008.08.006. Using this noncommutative deformed instanton, we investigate the spinor zero modes of the Dirac operator in a noncommutative instanton background on noncommutative , and we modify the index of the Dirac operator on the noncommutative space slightly and show that the number of the zero mode of the Dirac operator is preserved under the noncommutative deformation. We prove the existence of the Green's function associated with instantons on noncommutative , as a smooth deformation of the commutative case. The feature of the zero modes of the Dirac operator and the Green's function derives noncommutative ADHM (Atiyah-Drinfeld-Hitchin-Manin) equations which coincide with the ones introduced by Nekrasov and Schwarz. We show a one-to-one correspondence between the instantons on noncommutative and ADHM data. An example of a noncommutative instanton and a spinor zero mode are also given.

Y.M. is supported by JSPS KAKENHI Grant Nos. 23340018 and 22654011, and A.S is supported by JSPS KAKENHI Grant No. 23540117. We would like to thank Steven Rosenberg for his through reading, helpful suggestions, and comments. We would like to express our gratitude to Hiroshi Umetsu and Toshiya Suzuki for fruitful discussions and suggestions over a long period of time. We would like to thank Anca Tureanu, Masud Chaichian, and Claus Montonen for many important suggestions and comments. We would like to give special thanks to Masashi Hamanaka for his through reading our article and his comments, and for access to his work, which enabled us to complete this article. The authors appreciate for the referee's helpful comments.

Article outline:I. INTRODUCTIONII. NOTATIONS, DEFINITIONS, AND KNOWN FACTSIII. THE INDEX OF THE DIRAC OPERATORIV. GREEN'S FUNCTIONV. FROM INSTANTONS TO THE ADHM EQUATIONSVI. EXAMPLEVII. CONCLUSION